Solution 6.

See text and/or instructor's solution manual.

Solution.  First, consider    and calculate it's partial derivatives  

                    ,    ,    ,    .  

Substitute the values into Laplace's equation and get   

                    

Therefore    is a harmonic function.  

Second, consider    and calculate it's partial derivatives  

                    ,    ,    ,    .  

Substitute the values into Laplace's equation and get   

                    

Therefore    is a harmonic function.  

Third, consider the product    

and calculate it's partial derivatives  

                    ,    ,    ,    .  

Substitute the values into Laplace's equation and get   

                    .  

Therefore    is not a harmonic function.  

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell